Strong ferromagnetic exchange interaction in the parent state of the ...

Strong ferromagnetic exchange interaction in the parent state of the superconductivity in BaFe2 S3 Meng Wang,1, ∗ S. J. Jin,2 Ming Yi,1 Yu Song,3 H. C. Jiang,4 W. L. Zhang,5 H. Q. Luo,5 A. D. Christianson,6 E. Bourret-Courchesne,7 D. H. Lee,1, 7 Dao-Xin Yao,2 and R. J. Birgeneau1, 7, 8

arXiv:1609.00465v1 [cond-mat.supr-con] 2 Sep 2016

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Department of Physics, University of California, Berkeley, California 94720, USA 2 School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 3 Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA 4 Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, California 94025, USA 5 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 6 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 7 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 8 Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA Inelastic neutron scattering measurements have been performed to investigate the spin waves of the quasi-one-dimensional antiferromagnetic ladder compound BaFe2 S3 , where a superconducting transition was observed under pressure [H. Takahashi et al., Nat. Mater. 14, 1008-1012 (2015); T. Yamauchi et al., Phys. Rev. Lett. 115, 246402 (2015)]. By fitting the spherically averaged experimental data collected on a powder sample to a Heisenberg Hamiltonian, we find that the onedimensional antiferromagnetic ladder exhibits a strong nearest neighbor ferromagnetic exchange interaction (SJR = −71 ± 4 meV) along the rung direction, an antiferromagnetic SJL = 49 ± 3 meV along the leg direction and a ferromagnetic SJ2 = −15 ± 2 meV along the diagonal direction. Our data demonstrate that the antiferromagnetic spin excitations are a common characteristic for the iron-based superconductors, while specific relative values for the exchange interactions do not appear to be unique for the parent states of the superconducting materials. PACS numbers: 75.30.Ds,75.30.Et,78.70.Nx

The mechanism of high temperature (HT C) superconductivity has been one of the most intensely investigated topics since the discovery of the copper-oxide superconductors[1]. Analogous to the role of phonons in promoting superconductivity in conventional superconductors, spin fluctuations have been viewed as a possible glue that is essential for the formation of cooper pairs in the HT C superconductors[2, 3]. It has been shown that the spin fluctuations in both copper and iron-based superconductors (FeSC) are intimately coupled with the superconductivity, specifically, the appearance of a spin resonance mode in the superconducting (SC) state, and the doping dependence of the spin fluctuations in the normal state[4, 5]. The spin fluctuations in a SC compound derive from the spin waves of its magnetically ordered parent compound. Measurements of the spin waves in the parent compound are essential to determine the nature of the spin fluctuations and, in turn, to elucidate their role in the HT C superconductors including the possibility that the spin fluctuations are the primary pairing mechanism. Recently, a SC transition up to 24 K has been observed in the quasi-one-dimensional (1D) ladder compound BaFe2 S3 under pressure in the range of 10 to 17 GPa[8, 9]. The obtained pressure dependent phase diagram (Fig. 1 (a)) resembles that of the 1D copper oxide laddered system Sr14−x Cax Cu24 O41 (Fig. 1 (b))[7, 10–

12] and the commonly observed doping dependent phase diagrams in the layered FeSC[13]. This suggests that BaFe2 S3 at ambient pressure is the parent state of the superconductivity discovered under pressure, and that the superconductivity likely has a common origin, possibly magnetic-fluctuation-mediated[14]. It has been suggested that the abrupt increase of the N´ eel temperature (TN ) as a function of pressure shown in Fig. 1 (a) is associated with a quantum phase transition due to the change of orbital occupancies under pressure[6]. BaFe2 S3 is isostructural with the 1D antiferromagnetic (AF) ladder compounds AFe2 Se3 (A = K, Rb, and Cs, space group: Cmcm, no. 63) and similar to the slightly distorted material BaFe2 Se3 (space group: P nma, no. 62), as shown in Fig. 1(c)[15–22]. The thermal activation gap in BaFe2 S3 (∼ 70meV ) [16] is the smallest among the Fe-based ladder compounds, and photoemission studies suggest that both localized and itinerant electrons coexist at room temeprature[23]. The FeX (X = Se, S, As, and P) tetrahedra are common among the 1D AF ladder and 2D stripe ordered materials[24–27]. However, in contrast to the FeX tetrahedra in the other 1D AF ladders[22], the moments of BaFe2 S3 are smaller (∼ 1.2µB /Fe) and aligned along the rung direction, as shown in Fig. 1 (d)[8], and the distance of the Fe-Fe bonds along the AF direction (leg) is shorter than that along the ferromagnetic (FM) direction (rung). Hence, the spin dynamics,

2 we ground 8 g of the single crystals into a powder for this experiment. Our INS experiment was carried out on the ARCS time-of-flight chopper spectrometer[28] at the Spallation Neutron Source, Oak Ridge National laboratory (SNS, ORNL). The powder sample was sealed in an aluminum can and loaded into a He top-loading refrigerator. The sample was measured with incident beam energies of Ei = 50, 150, and 250 meV at 5 K. The energy resolutions for these incident beams were ∆E =2.2, 7.0, and 13.3 meV, as determined by the full width at half maximum (FWHM) of the energy cuts at E = 0 meV.

FIG. 1: (a) Pressure dependence of the AF and superconducting transitions, and the moment sizes (inset panel) for BaFe2 S3 adopted from Ref. [6]. (b) Pressure dependence of the spin gap (∆s ) and superconducting transitions for the laddered compound Sr2 Ca12 Cu24 O41 adopted from Ref. [7]. (c) A sketch of the ladder structure of BaFe2 S3 . The cuboid indicates one unit cell. (d) One-dimensional edge-shared FeS tetrahedra in BaFe2 S3 . The red arrows represent the moment directions of irons. The JL , JR , J2 , J5 and J7 are the magnetic exchange interactions between the corresponding irons.

predominately governed by the geometry of the lattice, could be different in BaFe2 S3 . Accordingly, it is important to measure the spin waves of BaFe2 S3 and extract the exchange interactions in order to compare with the other 1D and 2D analogs. In this paper, we report inelastic neutron scattering (INS) studies on the spin waves of a BaFe2 S3 powder sample. Similar to our measurements on RbFe2 Se3 [22], we observe an acoustic branch and an optical branch of spin waves, consistent with two inequivalent irons in the magnetic Brillouin zone. From the spherically averaged spectra on the powder sample, we are able to extract a spin gap, two band tops of the acoustic branch along two directions, and the minimum and maximum energies of the optical branch. By solving the Heisenberg Hamiltonian of the ladder structure with the observed constraints, we determine a set of parameters (SJR = −71 ± 4, SJL = 49 ± 3, SJ2 = −15 ± 2, SJ7 = 3.0 ± 0.5, and SJs = 0.1 ± 0.04 meV) with a strong intraladder FM exchange interaction along the rung direction that fits the experimental data well. The results demonstrate that the spin fluctuations are comparable among various parent compounds of the FeSC, while the exchange interactions that are previously proven universal are not unique for the stripe AF ordered parent state of the FeSC. The BaFe2 S3 samples were grown using the Bridgman method[27]; they formed in small needle-like single crystals, making them extremely difficult to align. Hence

Figure 2 shows INS spectra and cuts for the BaFe2 S3 powder samples with different incident energies. In Fig. 2(a), we can see intense excitations at Q = 1.27 ˚ A−1 , dispersive excitations stemming from Q = 2.19 and 2.81 ˚ A−1 , weak excitations at Q = 3.59 ˚ A−1 and a gap around 5 meV for all the Qs. The spectrum resembles the spin waves observed on the ladder compound RbFe2 Se3 [22]. The four Qs are consistent with the AF wave vectors at (H, K, L) = (0.5, 0.5, 1), (2.5, 0.5, 1), (3.5, 0.5, 1), and (0.5, 0.5, 3), revealing that the excitations are the spin waves of BaFe2 S3 . Here, (H, K, L) are p Miller indices for the momentum transfer |Q| = 2π (H/a)2 + (K/b)2 + (L/c)2 , where the lattice constants are a = 8.79, b = 11.23, and c = 5.29 ˚ A[8]. The flat excitations with intensities increasing with Q below 30 meV are phonons associated with the sample and the thin aluminum can. To determine the spin gap and dispersion relations quantitatively, we present constant Q cut integrated within Q = 1.27 ± 0.1 ˚ A−1 in Fig. 2(e) and constant energy cuts within E = 6 ± 1, 12 ± 1, 18 ± 1, 24 ± 1, and 30 ± 1 meV in Fig. 2(f). The minimum of the inladder plane and out-of-ladder plane spin gaps is 5 ± 1 meV[Supplementary materials]. The spin excitations stemming from Q = 2.19 and 2.81 ˚ A−1 disperse separately into four peaks with increasing energy. At around 30 meV, the two inner peaks merge together, indicating that the spin waves have reached a maximum along the [H, 0.5, 1] direction. Figures 2(b) and 2(g) present the dispersive spin excitations at Q = 3.59 ˚ A−1 with Ei = 80 meV. The spin excitations continuously evolve into dispersionless excitations at 70 meV, as shown in Figs. 2(c) and 2(h). This energy (∼ 70 meV) is higher than the cut-off energy of phonons and the intensities decrease with inceasing Q, indicating that they are magnetic excitations of BaFe2 S3 . The dispersion relation at Q = (0.5, 0.5, 3) = 3.59 ˚ A−1 corresponds to the dispersion along the [0.5, 0.5, L] direction. Thus, the dispersionless spin excitations at 70 meV can be ascribed to the zone boundary excitations along the L direction. Gaussian peak fittings to the constant Q cuts of the dispersionless spin excitations in Fig. 2(h) show centers at 71 ∼ 72 meV. The energy is significant lower than the observed spin wave maximum (∼ 190 meV) along the same direc-

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FIG. 2: (a) INS spectra S(Q, ω) of BaFe2 S3 at 5 K with Ei = 50, (b) 80, (c) 150, and (d) 250 meV. The color represents intensities in arbitrary units. The red dashed rectangles highlight the areas for the cuts in (g) and (h). (e) Constant Q cut with Ei = 50 meV between 1.26 < Q < 1.28 ˚ A−1 . (f) Constant energy cuts at E = 6, 12, 18, 24, and 30 meV integrated within E ± 1 meV with Ei = 50 meV. (g) Similar constant energy cuts at E = 6 ± 1, 35 ± 1 meV with Ei = 50 meV and E = 48 ± 1.5, 53 ± 1.5 meV with Ei = 80meV at 5 K. The dashed lines are guides to the dispersion relations of spin excitations. The solid line on top of E = 24 meV data points is a fit to Gaussian functions. The intensities for E = 24, 48, and 53 meV have been doubled for comparison. (h) Constant Q cuts at Q = 3.0, 3.7, 4.5 ˚ A−1 integrated within Q ± 0.15 ˚ A−1 with Ei = 150 meV and Q = 5.75, −1 −1 ˚ ˚ 7.75 A integrated within Q ± 0.25 A with Ei = 250 meV. The green solid lines are fits to Gaussian functions. The error bars are one standard deviation of the measured counts.

tion for RbFe2 Se3 [22]. In Fig. 2(d), we present the optical spin waves measured with Ei = 250 meV at 5 K. Two flat branches of excitations are observed. The center of the lower branch is determined to be at 171.6 ± 0.3 meV within Q = 5.75 ± 0.25 ˚ A−1 and 176 ± 2 meV within Q = 7.75 ± 0.25 −1 ˚ A , and that of the higher branch is at 210.7 ± 0.3 meV within Q = 7.75 ± 0.25 ˚ A−1 . The low and high branches of magnetic excitations are consistent with them being the minimum and maximum of the optical branch of the spin waves of BaFe2 S3 . The extracted spin wave dispersion relations have been plotted in Fig. 3. BaFe2 S3 at ambient pressure exhibits a stripe ordered structure similar to that of RbFe2 Se3 [22]. We proceed to employ the Heisenberg Hamiltonian that has been used to describe the spin waves of the ladder compound RbFe2 Se3 and other 2D stripe systems to fit the dispersion relations and extract the magnetic exchange interactions for BaFe2 S3 [22, 29–33]. The spin Hamiltonian is written as X Jr,r0 X ˆ = H Sr · Sr0 − Js (Szr )2 , (1) 2 0 r r,r

where Jr,r0 are the effective exchange couplings and (r, r0 ) label the iron sites, Js is the single ion Ising anisotropy term[34]. By solving Eq. 1 using the linear spin wave

approximation, the dispersion relations and extrema values can be obtained[34]. Because we have assumed identical Hamiltonians for the spin waves of BaFe2 S3 and RbFe2 Se3 , the solutions have the same analytical expressions[22]. The spin gap ∆s , the tops of the acousH tic mode along the H direction (E1t ) and L direction L (E1t ), and the bottom (E2b ) and top (E2t ) of the optical mode are as follows: p ∆s = 2S Js (2JL + 2J2 + J7 + Js ), p H E1t = 2S (2JL + 2J2 + Js )(J7 + Js ), p L E1t = 2S (JL + J2 + Js )(JL + J2 + J7 + Js ), p E2b = 2S (2JL − JR + Js )(2J2 − JR + J7 + Js ), p E2t = 2S (JL − JR + J2 + Js )(JL − JR + J2 + J7 + Js ). (2) The JR , JL and J2 are the intraladder exchange interactions along the rung, leg, and diagonal directions, respectively. J7 is the seventh nearest neighbor (NN) exchange interaction of irons between two ladders, as defined in Fig. 1(c). The expressions in Eq. (2) correspond to the wave vectors at Q = (H, L) = (0.5, 1), (1, 1), (0.5, 0.5), (1, 1), and (1, 0.5), respectively. The K for these wave vectors is 0.5.

4 TABLE I: The magnetic exchange couplings and NN Fe-Fe distances along the antiferromagnetic (JAF and dAF ) and ferromagnetic (JF and dF ) direction, respectively, and the exchange couplings along the diagonal direction for various Fe-based materials[22, 29–32]. The bond distances, dAF and dF , are in unites of angstrom (˚ A). Compounds CaFe2 As2 BaFe2 As2 SrFe2 As2 Rb2 Fe3 S4 RbFe2 Se3 BaFe2 S3

FIG. 3: Comparisons between the SpinW simulated spin excitation spectra and experimental determined dispersion relations (white points) for BaFe2 S3 . (a) Instrumental resolutions of 13.3 meV and (b) 5 meV have been convolved for comparison with the experimental data in Figs. 2(a)-2(d). (c) SpinW simulated spin excitations along high symmetry directions in the [H, L] 2D Brillouin zone for the parameters labeled on the figures. The other parameters, SJ7 and SJs , have been fixed at 3.0 and 0.1 meV, respectively. The color represents intensities. We convolve a constant 5 meV instrumental resolution for visualization. The inset in panel (c) shows a tetrahedron and associated exchange interactions.

From the spherically averaged INS data, we have deterH mined the values for these extrema, where ∆s ≈ 5, E1t ≈ L 30, E1t ≈ 72, E2b ≈ 172, and E2t ≈ 211 meV. Solving Eq. (2) would lead to two sets of mathematical solutions. By comparing with the experimental data, the two sets of parameters are determined as SJL = 49.3, SJ2 = −15.1 meV and SJL = −14.3, SJ2 = 48.4 meV, respectively, while the other interactions, SJR = −70.5, SJ7 = 3.0, and SJs = 0.1 meV, are the same. The two sets of parameters fit our spherically averaged data equally well. However, there is a difference for the optical spin wave branch for single crystals[Supplementary materials]. The intensity distribution of the optical mode for the second set of parameters disagrees with that of RbFe2 Se3 , where the intensities at (H, L) = (1, 1) are stronger than

SJAF 50 ± 10 59 ± 2 39 ± 2 42 ± 5 70 ± 5 49 ± 3

SJF SJ2 (meV) −6 ± 5 19 ± 4 −9 ± 2 14 ± 1 −5 ± 5 27 ± 1 −20 ± 2 17 ± 2 −12 ± 2 25 ± 5 −71 ± 4 −15 ± 1

dAF 2.753 2.808 2.785 2.76 2.77 2.64

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